SS3657 Lecture 5
I
Review of the basic properties of cdf functions.
I
The properties of pdf and cdf of a continuous r.v.
I
Textbook coverage: Chap 2.2
1/9
Identify which of the following plots display
University of Western Ontario Statistics 3657 Term Test I
October 19, 2009, 5:30 PM - 6:30 PM
Instructor: R. J. Kulperger
1 2 3 4 Total (50)
Name : ID :
1
Statistics 3657a Term Test, October 19, 2009,
University of Western Ontario Statistics 357 Term Test I
October 15, 2007, 5:30 PM - 6:30 PM
Instructor: R. J. Kulperger Instructions: I. Make sure that your name and ID number are the front page, and
University of Western Ontario Statistics 3657 Term Test II
22 November, 2010
Instructor: R. J. Kulperger Instructions: I. Make sure that your name and ID number are the front page, and every page, of
University of Western Ontario Statistics 357 Term Test II
12 November, 2007
Instructor: R. J. Kulperger Instructions: I. Make sure that your name and ID number are the front page, and every page, of y
Sample Mean and Variance for I.I.D. Normals
Suppose that Xi , i = 1, . . . , n are i.i.d. N (0, 1) random variables, n 2. The sample mean and variance are then given by X s2 = = 1 Xi n i=1
n
) 1 ( 2 X
Statistics 3657 : Moment Approximations
1
Preliminaries
Suppose that we have a r.v. X and that we wish to calculate the expectation of g(X) for some function g. Of course we could calculate it as E(g(
Statistics 3657 : Moment Generating Functions
A useful tool for studying sums of independent random variables is generating functions. In this course we consider moment generating functions. Definitio
Order Statistics and Distributions
1
Some Preliminary Comments and Ideas
In this section we consider a random sample X1 , X2 , . . . , Xn common continuous distribution function F and probability dens
Ratio of Two Independent Unif(-1,1) Random Variables
1
General Comments
Y Suppose that X, Y are iid Uniform(-1,1) random variables. Let Z = X . In this handout we find the pdf of Z using two methods.
SS3657 Lecture 1
Review old concepts with new perspective:
I Examples
I
I
The throwing two dices for a sum of 7;
The birthday problem
I
The concept of probability triplet (, F, P)
I
The axioms of prob
SS3657 Lecture 2
It is all about conditional probability
I Examples
I
I
The Polyas Urn model
The Monty hall problem
I
The definition of conditional probability
I
The Law of Total Probability
I
The Bay
SS3657 Lecture 3
I
The concept of independence
I
I
I
I
Examples
I
I
I
Pairwise independence v.s. (mutually) independence
Physical independence v.s. statistical independence
Mutually exclusive events a
SS3657 Lecture 4
I
The definition of random variables
I
The properties of cdf F ()
Discrete r.v.s
I
I
I
cdf and pmf of a discrete r.v.
Textbook coverage: Chap 2.1
1/17
The definition of random variabl
SS3657 Lecture 1
Review old concepts with new perspective:
I Examples in the form of clicker questions
I
I
The probability of obtaining a sum of 7 when throwing two
dices
The birthday problem
I
The co
SS3657 Lecture 3
I
I
Examples in the form of clicker questions: circuit example.
The concept of independence
I
I
I
I
Pairwise independence v.s. (mutually) independence
Physical independence v.s. stati
SS3657 Lecture 6
I
Transformations or functions of a r.v.
I
Textbook coverage: Chap 2.3
1/14
Transformation Y = g(X)
I
Consider a transformation Y = g(X), where g is a function
g : D 7 R.
The domain D
Urn Game Binary Tree Plots
Choose balls from an urn. The urn has R = 4 red balls and W = 6 white balls. Figure 1 shows the probability rules for a game without replacement and Figure 2 shows the proba
Change of Variables and Marginals: Example of Student's t Ratio
Suppose that X and Y are independent r.v.'s, X R and Y > 0. A specific example below is X N (0, 1) and Y 2 . (n) Consider the transforma
Law of Large Numbers and Central Limit Theorem
1
Convergence in Probability and Law of Large Numbers
Definition 1 A sequence of r.v.s Xn , n = 1, 2, 3, . . . is said to converge to Y if and only if fo
SOME HELPFUL FORMULAE 1. Gamma function and properties: (a) () =
0
x-1 e-x dx
(b) For > 0, ( + 1) = (). (c) For integers n 1, (n) = (n - 1)!. (d) (1/2) = 2. Binomial, parameter and n the number of tri
Chapter 3.3 Continuous RV and 3.4 Independent Random Variables
1
Continuous Multivariate Distributions
As in the discrete case we have a CDF for the d random variables X1 , . . . , Xn . It is a functi
Chapter 2.3 Transformation Examples
1
General Comments on Transformations and Distributions
In these examples X is a continuous random variable. In a previous handout we considered X as a discrete r.v
Statistics 357 : Axioms of Probability
Probability F : : set of possible outcomes (sample space) set of possible events
P : a rule or function that assigns probability to each event In finite outcomes
Statistics 3657 : Independent Events
Recall the definition of independence from class. Definition of independence of events 1. Events A1 , A2 , . . . , An are said to be be independent if and only if
Bivariate and Multivariate Normal
1
Bivariate Normal
cfw_ exp )(
1 - 2(1-2 ) y-Y Y
The bivariate normal pdf is given by f (x, y) =
1 2X Y 1-2
[(
-2
2 2 It has 5 parameters X , Y , X , Y , .
(
x-X X
)]
Statistics 3657a Assignment 5
Handout date: 3 December, 2011 Due date: At the start of the exam December 12, 2011
Problems from the text section 5.4 : 1, 3, 4, 7, 12, 20, 25 Hint : for 5.4.1 use Cheby